Its a huge solution, and so is the answer..

Answer is :

 2 * pi *(b-a) * h * (2a + b)/6 ....if you take density as 1...

I'll tell you how to do it..

Consider an element of d liquid at a depth x below surface.Let thickness of this be dx.

(Just like in moment of inertia derivations)

Find area of the element(dA) ....consider the trapezium as a rectangle..you get 2pi*r *dx/cos@...

@ is inclination of bucket wall...

Now find thrust on curved surface by this element usinf dF = dA * x*g(* d)...........where d is density..

Now integrate from 0 to h...to get DOWNWARD thrust..ie F = <integral > dF sin@.     ........sin@ since u need downward thrust..

Till here its straightforward..here you run into trouble.. 

u get a tan@ term...draw the figure and substitute values for that and r...

Solve and you get the answer..